Skip to main content
SpringerLink
Log in
Book cover

Conference on Computability in Europe

CiE 2007: Computation and Logic in the Real World pp 791–798Cite as

The Complexity of Small Universal Turing Machines

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4497))

Abstract

We survey some work concerned with small universal Turing machines, cellular automata, and other simple models of computation. For example it has been an open question for some time as to whether the smallest known universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are efficient (polynomial time) simulators of Turing machines. These are some of the most intuitively simple computational devices and previously the best known simulations were exponentially slow. We discuss recent work that shows that these machines are indeed efficient simulators. As a related result we also find that Rule 110, a well-known elementary cellular automaton, is also efficiently universal. We also mention some new universal program-size results, including new small universal Turing machines and new semi-weakly universal Turing machines. We then discuss some ideas for future work arising out of these, and other, results.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aaronson, S.: Book review: A new kind of science. Quantum Information and Computation 2(5), 410–423 (2002)

    Google Scholar 

  2. Baiocchi, C.: Three small universal Turing machines. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 1–10. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  3. Cocke, J., Minsky, M.: Universality of tag systems with P= 2. Journal of the Association for Computing Machinery 11(1), 15–20 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  4. Codd, E.: Cellular Automata. Academic Press, New York (1968)

    MATH  Google Scholar 

  5. Cook, M.: Universality in elementary cellular automata. Complex Systems 15(1), 1–40 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to parallel computation: P-completeness theory. Oxford university Press, Oxford (1995)

    MATH  Google Scholar 

  7. Harju, T., Margenstern, M.: Splicing systems for universal Turing machines. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA Computing. LNCS, vol. 3384, pp. 149–158. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  8. Hermann, G.T.: The uniform halting problem for generalized one state Turing machines. In: Proceedings of the ninth annual Symposium on Switching and Automata Theory (FOCS), October, pp. 368–372. IEEE Computer Society Press, Schenectady, New York (1968)

    Google Scholar 

  9. Ikeno, N.: A 6-symbol 10-state universal Turing machine. In: Proceedings, Institute of Electrical Communications, Tokyo (1958)

    Google Scholar 

  10. Kudlek, M.: Small deterministic Turing machines. Theoretical Computer Science 168(2), 241–255 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kudlek, M., Rogozhin, Y.: A universal Turing machine with 3 states and 9 symbols. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 311–318. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  12. Lindgren, K., Nordahl, M.G.: Universal computation in simple one-dimensional cellular automata. Complex Systems 4(3), 299–318 (1990)

    MathSciNet  MATH  Google Scholar 

  13. Margenstern, M.: Non-erasing Turing machines: A frontier between a decidable halting problem and universality. In: Ésik, Z. (ed.) FCT 1993. LNCS, vol. 710, pp. 375–385. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  14. Margenstern, M.: Non-erasing Turing machines: a new frontier between a decidable halting problem and universality. In: Baeza-Yates, R.A., Poblete, P.V., Goles, E. (eds.) LATIN 1995. LNCS, vol. 911, pp. 386–397. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  15. Margenstern, M.: The laterality problem for non-erasing Turing machines on {0, 1} is completely solved. Theoretical Informatics and Applications 31(2), 159–204 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Margenstern, M.: Frontier between decidability and undecidability: a survey. Theoretical Computer Science 231(2), 217–251 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Margenstern, M.: An algorithm for building intrinsically universal cellular automata in hyperbolic spaces. In: Proceedings of the 2006 International Conference on Foundations of Computer Science (FCS), pp. 3–9. CSREA Press, Las Vegas, NV (2006)

    Google Scholar 

  18. Margenstern, M., Pavlotskaya, L.: On the optimal number of instructions for universality of Turing machines connected with a finite automaton. International Journal of Algebra and Computation 13(2), 133–202 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Michel, P.: Small Turing machines and generalized busy beaver competition. Theoretical Computer Science 326, 45–56 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Minsky, M.: A 6-symbol 7-state universal Turing machines. Technical Report 54-G-027, MIT (August 1960)

    Google Scholar 

  21. Minsky, M.: Size and structure of universal Turing machines using tag systems. In: Recursive Function Theory: Proceedings, Symposium in Pure. Mathematics, Provelence, AMS vol. 5, pp. 229–238 (1962)

    Google Scholar 

  22. Moore, C.: Quasi-linear cellular automata. Physica D 103, 100–132 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moore, C.: Predicting non-linear cellular automata quickly by decomposing them into linear ones. Physica D 111, 27–41 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Neary, T.: Small polynomial time universal Turing machines. In: MFCSIT’06. Fourth Irish Conference on the Mathematical Foundations of Computer Science and Information Technology, pp. 325–329. University College Cork, Ireland (2006)

    Google Scholar 

  25. Neary, T., Woods, D.: Four small universal Turing machines. In: Machines, computations and universality (MCU 2007) LNCS, September 2007. Springer (accepted)

    Google Scholar 

  26. Neary, T., Woods, D.: A small fast universal Turing machine. Technical Report NUIM-CS-TR-2005-12, Department of Computer Science, NUI Maynooth (2005)

    Google Scholar 

  27. Neary, T., Woods, D.: P-completeness of cellular automaton Rule 110. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 132–143. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  28. Neary, T., Woods, D.: Small fast universal Turing machines. Theoretical Computer Science 362(1–3), 171–195 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ollinger, N.: The quest for small universal cellular automata. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 318–329. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  30. Pavlotskaya, L.: Solvability of the halting problem for certain classes of Turing machines. Mathematical Notes (Springer), vol. 13(6) pp. 537–541, June 1973 (Translated from Matematicheskie Zametki, vol. 13, No. 6, pp. 899–909, June 1973)

    Google Scholar 

  31. Pavlotskaya, L.: Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T’juring. Avtomaty i Mashiny, pp. 91–118 (Sufficient conditions for the halting problem decidability of Turing machines) (in Russian) (1978)

    Google Scholar 

  32. Priese, L.: Towards a precise characterization of the complexity of universal and nonuniversal Turing machines. SIAM J. Comput. 8(4), 508–523 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  33. Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae 12(3), 177–209 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  34. Robinson, R.M.: Minsky’s small universal Turing machine. International Journal of Mathematics 2(5), 551–562 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rogozhin, Y.: Sem’ universal’nykh mashin T’juringa. Systems and theoretical programming, Mat. Issled (Seven universal Turing machines. In Russian) 69, 76–90 (1982)

    Google Scholar 

  36. Rogozhin, Y.: Small universal Turing machines. Theoretical Computer Science 168(2), 215–240 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rogozhin, Y., Verlan, S.: On the rule complexity of universal tissue P systems. In: Freund, R. (ed.) WMC 2005. LNCS, vol. 3850, pp. 356–362. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  38. Shannon, C.E.: A universal Turing machine with two internal states. Automata Studies, Annals of Mathematics Studies 34, 157–165 (1956)

    MathSciNet  Google Scholar 

  39. Siegelmann, H.T., Margenstern, M.: Nine switch-affine neurons suffice for Turing universality. Neural Networks 12(4–5), 593–600 (1999)

    Article  Google Scholar 

  40. Watanabe, S.: On a minimal universal Turing machines. Technical report, MCB Report, Tokyo (August 1960)

    Google Scholar 

  41. Watanabe, S.: 5-symbol 8-state and 5-symbol 6-state universal Turing machines. Journal of the ACM 8(4), 476–483 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  42. Watanabe, S.: 4-symbol 5-state universal Turing machines. Information Processing Society of Japan Magazine 13(9), 588–592 (1972)

    Google Scholar 

  43. Wolfram, S.: Statistical mechanics of cellular automata. Reviews of Modern Physics 55(3), 601–644 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  44. Woods, D., Neary, T.: Small semi-weakly universal Turing machines. In: Machines, computations and universality(MCU) 2007, Sept. Springer LNCS. (Accepted)

    Google Scholar 

  45. Woods, D., Neary, T.: On the time complexity of 2-tag systems and small universal Turing machines. In: 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), October, pp. 439–446. IEEE, Berkeley, California (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University College Cork, Ireland

    Damien Woods

  2. TASS, Department of Computer Science, National University of Ireland Maynooth, Ireland

    Turlough Neary

Authors
  1. Damien Woods
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Turlough Neary
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Pure Mathematics, University of Leeds, LS2 9JT, Leeds, UK

    S. Barry Cooper

  2. Institue for Logic, Language and Computation (ILLC), Universiteit van Amsterdam, Plantage Muidergracht 24, 1018, TV Amsterdam, The Netherlands

    Benedikt Löwe

  3. Dipartimento di Scienze Matematiche ed Informatiche “R. Magari”, University of Siena, 53100, Siena, Italy

    Andrea Sorbi

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Woods, D., Neary, T. (2007). The Complexity of Small Universal Turing Machines. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_84

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-73001-9_84

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73000-2

  • Online ISBN: 978-3-540-73001-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation